Appendix E — Problem Set 3 | COLLECTION

Companion to §1.2–1.3 — sets and counting

Use the distinction deliberately: a permutation counts ordered selections; a combination counts unordered selections.

A. Recognition

1. Does order matter?

For each problem, decide whether order matters and choose n^r, P(n,r), \binom nr, or a multinomial coefficient.

  1. Choose three samples from twelve for destructive analysis.
  2. Assign three distinct samples to instruments A, B, and C.
  3. Construct a five-residue peptide from twenty amino-acid types, allowing repetition.
  4. Distribute six isotopic labels among six fixed sites, using two ^{13}\mathrm C labels and four ^{12}\mathrm C labels.

2. COLLECTION or ARRANGEMENT?

For each pair, state whether it represents the same COLLECTION, the same ARRANGEMENT, both, or neither.

  1. \{\mathrm H,\mathrm H,\mathrm O\} and \{\mathrm O,\mathrm H,\mathrm H\} as bags of atoms.
  2. H–O–H and H–H–O as bonded sequences.
  3. cis- and trans-[\mathrm{PtCl_2(NH_3)_2}].
  4. two electron configurations that differ only by exchanging indistinguishable electrons.

B. Counting tools

3. Derive, do not quote

Derive \binom nr=n!/[r!(n-r)!] by first counting ordered selections and then identifying the overcount. Evaluate \binom{12}{4}.

4. Isotopologues of methane

Methane has four equivalent hydrogen sites. Suppose exactly two H atoms are replaced by D.

  1. How many site-labelings exist before molecular symmetry is considered?
  2. Why does this count not necessarily equal the number of experimentally distinguishable isotopomers?
  3. What additional mathematical object is required to account for molecular symmetry?

5. Peptide sequences

Assume the twenty standard amino acids are available.

  1. How many tripeptide sequences exist?
  2. How many contain no repeated residue type?
  3. How many contain exactly two alanines?
  4. Among all tripeptides, what fraction contain exactly two alanines?

6. Ligand placement

Ignore geometric symmetry initially. Six labeled coordination sites receive two A ligands, two B ligands, and two C ligands.

  1. Find the number of site assignments.
  2. Generalize to counts n_A,n_B,n_C with N=n_A+n_B+n_C.
  3. Explain precisely where indistinguishability enters the denominator.

C. Multiplicity and entropy

7. Two-state molecular ensemble

Ten distinguishable molecules can occupy a lower state L or upper state U. Exactly four occupy U.

  1. Calculate the multiplicity \Omega.
  2. Calculate S/k_\mathrm B=\ln\Omega.
  3. Repeat for five molecules in U.
  4. Which macrostate is more probable if all microstates are equally likely?

8. Why the logarithm is forced

Independent systems A and B have multiplicities \Omega_A=10^6 and \Omega_B=10^9.

  1. Calculate \Omega_{AB}.
  2. Show that \ln\Omega_{AB}=\ln\Omega_A+\ln\Omega_B.
  3. Explain why an extensive entropy cannot be directly proportional to \Omega.
  4. Find \Delta(S/k_\mathrm B) when multiplicity increases by a factor of 100.

9. Binomial composition

An ideal mixture contains N=20 distinguishable lattice sites, occupied by 8 A molecules and 12 B molecules.

  1. Calculate the configurational multiplicity.
  2. Calculate S_\mathrm{mix}/k_\mathrm B=\ln\Omega.
  3. Compare with the multiplicity at 10 A and 10 B.
  4. Explain the result without saying merely “the number is larger.”

10. Model audit

A student claims that ten coin tosses with five heads have 10^5 arrangements.

  1. Identify what 10^5 would actually count, if anything.
  2. Give the correct count.
  3. State a recognition question that would have prevented the error.